The Topological Entropy of Powers on Lie Groups

TL;DR

This paper computes the topological entropy of power maps on Lie groups, extending known results from commutative cases to non-commutative groups using Lie theory, revealing that such entropy is positive for compact groups with discrete center.

Contribution

It generalizes the formula for topological entropy of power maps from commutative Lie groups to all Lie groups, including non-commutative cases, using structure theory.

Findings

Topological entropy of powers on Lie groups equals the dimension of a maximal torus times log(k).

For compact Lie groups with discrete center, the entropy is always positive.

Non-commutative cases differ from endomorphisms, which have zero entropy.

Abstract

This article addresses the problem of computing the topological entropy of an application $\psi : G \to G$, where $G$ is a Lie group, given by some power $\psi(g) = g^k$, with $k$ a positive integer. When $G$ is commutative, $\psi$ is an endomorphism and its topological entropy is given by $h(\psi) = \dim(T(G)) \log(k)$, where $T(G)$ is the maximal torus of $G$, as shown in \cite{patrao:endomorfismos}. But when $G$ is not commutative, $\psi$ is no longer an endomorphism and these previous results cannot be used. Still, $\psi$ has some interesting symmetries, for example, it commutes with the conjugations of $G$. In this paper, the structure theory of Lie groups is used to show that $h(\psi) = \dim(T)\log(k)$, where $T$ is a maximal torus of $G$, generalizing the commutative case formula. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.